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In this work we use the low-energy EFT called potential non-relativistic QCD(pNRQCD) to calculate the partial decay width of different $b\bar{b}$-statesundergoing an electric dipole transition at next-to-next-to leading order(NNLO). Explicitly, the $\chi_{b J}$, for $J=0,1,2$, and the $h_b$ areinvestigated by computing the processes $\chi_{b J} \to \Upsilon + \gamma$ and$h_b \to \eta_b + \gamma$. Relativistic corrections of relative order $v^2$ tothe leading electric dipole operator are included. The analysis separates thosecontributions that account for the electromagnetic interaction terms in thepNRQCD Lagrangian, which are $v^2$ suppressed, and those that account forquarkonium state corrections of relative order $v$ and $v^2$. Within the lastones, corrections come from higher order potentials ($\frac{1}{m}$ and$\frac{1}{m^2}$ terms), and from higher order Fock states which account for thecoupling of the quark-antiquark state to other low-energy degrees of freedomand thus demand non-perturbative input. Finally, the experimentally knownbranching fractions are used to predict the total decay with of the respectiveinitial states.

Heavy quarkonium electric dipole transitions in non-relativistic quantum field theory